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Let A = F q [ T ] be the polynomial ring over F q , and F be the field of fractions of A. Let ϕ be a Drinfeld A-module of rank r ≥ 2 over F. For all but finitely many primes p ◁ A , one can reduce ϕ modulo p to obtain a Drinfeld A-module ϕ ⊗ F p of rank r over F p = A / p . The endomorphism ring E p = End F p ( ϕ ⊗ F p ) is an order in an imaginary field extension K of F of degree r. Let O p be the integral closure of A in K, and let π p ∈ E p be the Frobenius endomorphism of ϕ ⊗ F p . Then we have the inclusion of orders A [ π p ] ⊂ E p ⊂ O p in K. We prove that if End F alg ( ϕ ) = A , then for arbitrary non-zero ideals n , m of A there are infinitely many p such that n divides the index χ ( E p / A [ π p ] ) and m divides the index χ ( O p / E p ) . We show that the index χ ( E p / A [ π p ] ) is related to a reciprocity law for the extensions of F arising from the division points of ϕ. In the rank r = 2 case we describe an algorithm for computing the orders A [ π p ] ⊂ E p ⊂ O p , and give some computational data.

A fundamental result of Springer says that a quadratic form over a field of characteristic ≠ 2 is isotropic if it is so after an odd degree field extension. In this paper we generalize Springer’s theorem as follows. Let R be an arbitrary semilocal ring, let S be a finite R -algebra of constant odd degree, which is etale or generated by one element, and let q be a nonsingular R -quadratic form whose base ring extension q S is isotropic. We show that then already q is isotropic.

We present a complete algorithm that computes all hypergeometric solutions of homogeneous linear difference equations and rational solutions of parameterized linear difference equations in the setting of Π Σ ⁎ -fields. More generally, we provide a flexible framework for a big class of difference fields that are built by a tower of Π Σ ⁎ -field extensions over a difference field that enjoys certain algorithmic properties. As a consequence one can compute all solutions in terms of indefinite nested sums and products that arise within the components of a parameterized linear difference equation, and one can find all hypergeometric solutions of a homogeneous linear difference equation that are defined over the arising sums and products.

Let G be a connected simple graph with characteristic polynomial P G ( x ) . The irreducibility of P G ( x ) over rational numbers Q has a close relationship with the automorphism group, reconstruction and controllability of a graph. In this paper we derive three methods to construct graphs with irreducible characteristic polynomials by appending paths P 2 n + 1 − 2 ( n ≥ 1 ) to certain vertices; union and join K 1 alternately and corona. These methods are based on Eisenstein's criterion and field extensions. Concrete examples are also supplied to illustrate our results.

Let K be a maximal unramified extension of a nonarchimedean local field with arbitrary residual characteristic p. Let G be a reductive group over K which splits over a tamely ramified extension of K. We show that the associated Moy-Prasad filtration representations are in a certain sense independent of p. We also establish descriptions of these representations in terms of explicit Weyl modules and as representations occurring in a generalized Vinberg-Levy theory. As an application, we use these results to provide necessary and sufficient conditions for the existence of stable vectors in Moy-Prasad filtration representations, which extend earlier results by Reeder and Yu (which required p to be large) and by Romano and the author (which required G to be absolutely simple and split). This yields new supercuspidal representations. We also treat reductive groups G that are not necessarily split over a tamely ramified field extension.

For a given field F of characteristic different from 2 and $$a,b,d\in F^*$$ we construct an invariant $$\mathrm{inv}$$ for an element $$D\in \,_2\mathrm{Br}(F(\sqrt{a},\sqrt{b},\sqrt{d})/F)$$ . This invariant takes value in the quotient group $$\begin{aligned} H^3(F,\mu _2)/D\cup {\mathrm{N}_{\mathrm{F}\left( \sqrt{\mathrm{d}}, \sqrt{\mathrm{ab}}\right) /\mathrm{F}}}F\left( \sqrt{d},\sqrt{ab}\right) ^*. \end{aligned}$$ Let k be a field, let $$k(\sqrt{a},\sqrt{b},\sqrt{d})/k$$ be a triquadratic field extension. We apply the invariant $$\mathrm{inv}$$ and a few deep results from algebraic geometry and K-theory to construct a field extension K/k with $$\mathrm{cd}_2 K=3$$ , and an indecomposable cross product algebra of exponent 2 with respect to the extension $$K(\sqrt{a},\sqrt{b},\sqrt{d})/K$$ . Using the invariant $$\mathrm{inv}$$ , we also prove the following odd degree descent statement: Assume $$D\in \,_2\mathrm{Br}(F)$$ , $$b,d\in F^*$$ , L/F is an odd degree extension. Assume also that $$D_{L(\sqrt{b},\sqrt{d})}=Q_{L(\sqrt{b},\sqrt{d})}$$ , where Q is a quaternion algebra defined over L. Then there exists a quaternion algebra $$\widetilde{Q}$$ defined over F such that $$D_{F(\sqrt{b},\sqrt{d})}=\widetilde{Q}_{F(\sqrt{b},\sqrt{d})}$$ . As a consequence we get that if $$\phi \in I^2(F)$$ is a form such that $${(\phi _{L(\sqrt{b},\sqrt{d})})}_{an}$$ is defined over L, and $$\dim {(\phi _{L(\sqrt{b},\sqrt{d})})}_{an} =4$$ , then $${(\phi _{F(\sqrt{b},\sqrt{d})})}_{an}$$ is defined over F.

It is an open problem whether the Iwasawa λ-invariants of the Z p -extensions of a fixed number field are bounded. Using the class-field theoretic tool of logarithmic class groups, we obtain bounds for the λ-invariants of Z p -extensions of suitable field extensions of imaginary quadratic number fields. We also prove the Gross-Kuz'min Conjecture for certain families of non-cyclotomic Z p -extensions.

This paper explores the existence and distribution of primitive elements in finite field extensions with prescribed traces in several intermediate field extensions. Our main result provides an inequality-like condition to ensure the existence of such elements. We then derive concrete existence results for a special class of intermediate extensions., Comment: This new version contains many corrections, and some minor results were removed. ** To appear in J. Aust. Math. Soc

Let $k$ be a field with char $k\neq 2$ and $k$ be not algebraically closed. Let $a\in k\setminus k^2$ and $L=k(\sqrt{a})(x,y)$ be a field extension of $k$ where $x,y$ are algebraically independent over $k$. Assume that $\sigma$ is a $k$-automorphism on $L$ defined by \[ \sigma: \sqrt{a}\mapsto -\sqrt{a},\ x\mapsto \frac{b}{x},\ y\mapsto \frac{c(x+\frac{b}{x})+d}{y} \] where $b,c,d \in k$, $b\neq 0$ and at least one of $c,d$ is non-zero. Let $L^{\langle\sigma\rangle}=\{u\in L:\sigma(u)=u\}$ be the fixed subfield of $L$. We show that $L^{\langle\sigma\rangle}$ is isomorphic to the function field of a certain surface in $P^4_k$ which is given as the intersection of two quadrics. We give criteria for the $k$-rationality of $L^{\langle\sigma\rangle}$ by using the Hilbert symbol. As an appendix of the paper, we also give an alternative geometric proof of a part of the result which is provided to the authors by J.-L. Colliot-Th\'el\`ene., Comment: To appear in Manuscripta Math. The main theorems (old Theorem 1.7 and Theorem 1.8) incorporated into (new) Theorem 1.8. Section 3 and Section 4 interchanged

Let K be a finitely generated extension of a field k of characteristic $$p\not =0$$ . In 1947, Dieudonne initiated the study of maximal separable intermediate fields. He gave in particular the form of an important subclass of maximal separable intermediate fields D characterized by the property $$K\subseteq k({D}^{p^{-\infty }})$$ , and which are called the distinguished subfields of K/k. In 1970, Kraft showed that the distinguished maximal separable subfields are precisely those over which K is of minimal degree. This paper grew out of an attempt to find a new characterization of distinguished subfields of K/k by means of new inseparability invariants.

A new way to formulate the notions of minimal basis and minimal indices is developed in this paper, based on the concept of a filtration of a vector space. The goal is to provide useful new tools for working with these important concepts, as well as to gain deeper insight into their fundamental nature. This approach also readily reveals a strong minimality property of minimal indices, from which follows a characterization of the vector polynomial bases in rational vector spaces. The effectiveness of this new formulation is further illustrated by proving several fundamental properties: the invariance of the minimal indices of a matrix polynomial under field extension, the direct sum property of minimal indices, the polynomial linear combination property, and the predictable degree property.

Over a global field any finite number of central simple algebras of exponent dividing m is split by a common cyclic field extension of degree m. We show that the same property holds for function fields of 2-dimensional excellent schemes over a henselian local domain of dimension one or two with algebraically closed residue field.

This paper is a continuation of our previous works where we study maps from $X_0(N)$, $N \ge 1$, into $\mathbb P^2$ constructed via modular forms of the same weight and criteria that such a map is birational (see [12]). In the present paper our approach is based on the theory of primitive elements in finite separable field extensions. We prove that in most of the cases the constructed maps are birational, and we consider those such that the resulting equation of the image in $\mathbb P^2$ is simplest possible., Comment: arXiv admin note: text overlap with arXiv:1305.2428

The relation between soft element-wise field and soft int-field has been established and then some properties of soft int-field are studied. We define the notions of soft algebraic element and soft purely inseparable element of a soft int-field extension. Some characterizations of soft algebraic and soft purely inseparable int-field extensions are given. Lastly, we define soft separable algebraic int-field extension and study some of its properties.

We initiate the study of affine Deligne–Lusztig varieties with arbitrarily deep level structure for general reductive groups over local fields. We prove that for $${{\,\mathrm{GL}\,}}_n$$ and its inner forms, Lusztig’s semi-infinite Deligne–Lusztig construction is isomorphic to an affine Deligne–Lusztig variety at infinite level. We prove that their homology groups give geometric realizations of the local Langlands and Jacquet–Langlands correspondences in the setting that the Weil parameter is induced from a character of an unramified field extension. In particular, we resolve Lusztig’s 1979 conjecture in this setting for minimal admissible characters.

Let k be a field of characteristic p≠ 0. In 1968, M. E. Sweedler revealed for the first time, the usefulness of the concept of modularity. This notion, which plays an important role especially for Galois theory of purely inseparable extensions, was used to characterize purely inseparable extensions of bounded exponent which were tensor products of simple extensions. A natural extension of the definition of modularity is to say that K/k is q-modular (quasi-modular) if K is modular up to some finite extension. In subsequent papers, M. Chellali and the author have studied various property of q-modular field extensions, including the questions of q-modularity preservation in case [k : kp] is finite. This paper grew out of an attempt to find analogue results concerning the preservation of q-modularity, without the hypothesis on k but with extra assumptions on K/k. In particular, we investigate existence conditions of lower (resp. upper) quasi-modular closures for a given q-finite extension.

We determine the Hopf Galois structures on a Galois field extension of degree twice an odd prime square and classify the corresponding skew left braces. Besides we determine the separable field extensions of degree twice an odd prime square allowing a cyclic Hopf Galois structure and the number of these structures.

Given a collection A={L1,…,Ln} of intermediate fields in a field extension L/K of finite degree and ΛL,A=L1×⋯×Ln, there is a natural map ΨL,A:L→ΛL,A given by y↦(TrL/L1(y),…,TrL/Ln(y)), where TrL/Li...

The objective of the study is to find out the ability of the field extension officers to implement communicative extension teaching technique, knowledge of extension, extension score, and creative attitude. The study was conducted in Jambi Extension Center (2019) with 90 respondents randomly selected. The study shows that there are positive correlations between: 1) knowledge of extension and the ability of the field extension officers to implement communicative extension teaching technique; 2) extension score and the ability of the field extension officers to implement communicative extension teaching technique; 3) creative attitude and the ability of the field extension officers to implement communicative extension teaching technique. In addition, there is a positive correlation between knowledge of extension, extension score and creative attitude & the ability of the field extension officers to implement communicative extension teaching technique Thus, the ability of the field extension officers to implement communicative extension teaching technique can be enhanced by increasing knowledge of extension, extension score, and creative attitude.

We study q-ary linear codes C obtained from Veronese surfaces over finite fields. We show how one can find the higher weight spectra of these codes, or equivalently, the weight distribution of all extension codes of C over all field extensions of the field with q elements. Our methods will be a study of the Stanley-Reisner rings of a series of matroids associated to each code C, 14 pages

Let K be a number field, and let G be a finitely generated and torsion-free subgroup of K × . We consider Kummer extensions of G of the form K ( ζ 2 m , G 2 n ) / K ( ζ 2 m ) , where n ⩽ m . In the paper by Debry and Perucca (2016) [1] , the degrees of those extensions have been evaluated in terms of divisibility parameters over K ( ζ 4 ) . We prove how properties of G over K explicitly determine the divisibility parameters over K ( ζ 4 ) . This result yields a clear computational advantage, since no field extension is required.

We study the maximal multiplicity locus of a variety X over a field of characteristic p>0 that is provided with a finite surjective radicial morphism δ:X→V, where V is regular, for example, when X⊂An+1 is a hypersurface defined by an equation of the form Tq−f(x1,…,xn)=0 and δ is the projection onto V:=Spec(k[x1,…,xn]). The multiplicity along points of X is bounded by the degree, say d, of the field extension K(V)⊂K(X). We denote by Fd(X)⊂X the set of points of multiplicity d. Our guiding line is the search for invariants of singularities x∈Fd(X) with a good behavior property under blowups X′→X along regular centers included in Fd(X), which we call invariants with the pointwise inequality property. A finite radicial morphism δ:X→V as above will be expressed in terms of an OVq-submodule M⊆OV. A blowup X′→X along a regular equimultiple center included in Fd(X) induces a blowup V′→V along a regular center and a finite morphism δ′:X′→V′. A notion of transform of the OVq-module M⊂OV to an OV′q-module M′⊂OV′ will be defined in such a way that δ′:X′→V′ is the radicial morphism defined by M′. Our search for invariants relies on techniques involving differential operators on regular varieties and also on logarithmic differential operators. Indeed, the different invariants we introduce and the stratification they define will be expressed in terms of ideals obtained by evaluating differential operators of V on OVq-submodules M⊂OV.

Consider the abelian category ${\mathcal C}$ of commutative group schemes of finite type over a field $k$, its full subcategory ${\mathcal F}$ of finite group schemes, and the associated pro category ${\rm Pro}({\mathcal C})$ (resp. ${\rm Pro}({\mathcal F})$) of pro-algebraic (resp. profinite) group schemes. When $k$ is perfect, we show that the profinite fundamental group $\varpi_1 : {\rm Pro}({\mathcal C}) \to {\rm Pro}({\mathcal F})$ is left exact and commutes with base change under algebraic field extensions; as a consequence, the higher profinite homotopy functors $\varpi_i$ vanish for $i \geq 2$. Along the way, we describe the indecomposable projective objects of ${\rm Pro}({\mathcal C})$ over an arbitrary field $k$., Final version, accepted for publication at Annales Henri Lebesgue

This article investigates the cohomological kernels ker(H2n+1F→H2n+1E) of field extensions E/F in characteristic two where separable part of E is a quadratic extension F(α) and E is quadratic or qu...

Every Lie algebra over a field E gives rise to new Lie algebras over any subfield F ⊆ E by restricting the scalar multiplication. This paper studies the structure of these underlying Lie algebra in relation to the structure of the original Lie algebra, in particular the question how much of the original Lie algebra can be recovered from its underlying Lie algebra over subfields F. By introducing the conjugate of a Lie algebra we show that in some specific cases the Lie algebra is completely determined by its underlying Lie algebra. Furthermore we construct examples showing that these assumptions are necessary. As an application, we give for every positive n an example of a real 2-step nilpotent Lie algebra which has exactly n different bi-invariant complex structures. This answers an open question by Di Scala, Lauret and Vezzoni motivated by their work on quasi-Kahler Chern-flat manifolds in differential geometry. The methods we develop work for general Lie algebras and for general Galois extensions F ⊆ E , in contrast to the original question which only considered nilpotent Lie algebras of nilpotency class 2 and the field extension R ⊆ C . We demonstrate this increased generality by characterizing the complex Lie algebras of dimension ≤4 which are defined over R and over Q .

Summary This is the third part of a four-article series containing a Mizar [3], [1], [2] formalization of Kronecker’s construction about roots of polynomials in field extensions, i.e. that for every field F and every polynomial p ∈ F [X]\F there exists a field extension E of F such that p has a root over E. The formalization follows Kronecker’s classical proof using F [X]/ as the desired field extension E [6], [4], [5]. In the first part we show that an irreducible polynomial p ∈ F [X]\F has a root over F [X]/. Note, however, that this statement cannot be true in a rigid formal sense: We do not have F ⊆ F [X]/ < p > as sets, so F is not a subfield of F [X]/, and hence formally p is not even a polynomial over F [X]/ < p >. Consequently, we translate p along the canonical monomorphism ϕ: F → F [X]/ and show that the translated polynomial ϕ (p) has a root over F [X]/. Because F is not a subfield of F [X]/ we construct in the second part the field (E \ ϕF)∪F for a given monomorphism ϕ: F → E and show that this field both is isomorphic to F and includes F as a subfield. In the literature this part of the proof usually consists of saying that “one can identify F with its image ϕF in F [X]/ and therefore consider F as a subfield of F [X]/”. Interestingly, to do so we need to assume that F ∩ E = ∅, in particular Kronecker’s construction can be formalized for fields F with F ∩ F [X] = ∅. Surprisingly, as we show in this third part, this condition is not automatically true for arbitrary fields F : With the exception of ℤ2 we construct for every field F an isomorphic copy F′ of F with F′ ∩ F′ [X] ≠ ∅. We also prove that for Mizar’s representations of ℤ n , ℚ and ℝ we have ℤ n ∩ ℤ n [X] = ∅, ℚ ∩ ℚ[X] = ∅ and ℝ ∩ ℝ[X] = ∅, respectively. In the fourth part we finally define field extensions: E is a field extension of F iff F is a subfield of E. Note, that in this case we have F ⊆ E as sets, and thus a polynomial p over F is also a polynomial over E. We then apply the construction of the second part to F [X]/ with the canonical monomorphism ϕ: F → F [X]/. Together with the first part this gives – for fields F with F ∩ F [X] = ∅ – a field extension E of F in which p ∈ F [X]\F has a root.

We prove a new effective Chebotarev density theorem for Galois extensions $L/\mathbb{Q}$ that allows one to count small primes (even as small as an arbitrarily small power of the discriminant of $L$); this theorem holds for the Galois closures of "almost all" number fields that lie in an appropriate family of field extensions. Previously, applying Chebotarev in such small ranges required assuming the Generalized Riemann Hypothesis. The error term in this new Chebotarev density theorem also avoids the effect of an exceptional zero of the Dedekind zeta function of $L$, without assuming GRH. We give many different "appropriate families," including families of arbitrarily large degree. To do this, we first prove a new effective Chebotarev density theorem that requires a zero-free region of the Dedekind zeta function. Then we prove that almost all number fields in our families yield such a zero-free region. The innovation that allows us to achieve this is a delicate new method for controlling zeroes of certain families of non-cuspidal $L$-functions. This builds on, and greatly generalizes the applicability of, work of Kowalski and Michel on the average density of zeroes of a family of cuspidal $L$-functions. A surprising feature of this new method, which we expect will have independent interest, is that we control the number of zeroes in the family of $L$-functions by bounding the number of certain associated fields with fixed discriminant. As an application of the new Chebotarev density theorem, we prove the first nontrivial upper bounds for $\ell$-torsion in class groups, for all integers $\ell \geq 1$, applicable to infinite families of fields of arbitrarily large degree., Comment: 52 pages. This shorter version aligns with the published paper. Note that portions of Section 8 of the longer v1 have been developed as a separate paper with identifier arXiv:1902.02008

A theorem of Chow concerns homomorphisms of two abelian varieties under a primary field extension base change. In this paper, we generalize Chow’s theorem to semi-abelian varieties. This contributes to different proofs of a well-known result that every algebraic torus splits over a finite separable field extension. We also obtain the best bound for the degrees of splitting fields of tori.

Summary This is the first part of a four-article series containing a Mizar [3], [1], [2] formalization of Kronecker’s construction about roots of polynomials in field extensions, i.e. that for every field F and every polynomial p ∈ F [X]\F there exists a field extension E of F such that p has a root over E. The formalization follows Kronecker’s classical proof using F [X]/ as the desired field extension E [9], [4], [6]. In this first part we show that an irreducible polynomial p ∈ F [X]\F has a root over F [X]/. Note, however, that this statement cannot be true in a rigid formal sense: We do not have F ⊆ [X]/ < p > as sets, so F is not a subfield of F [X]/, and hence formally p is not even a polynomial over F [X]/ < p >. Consequently, we translate p along the canonical monomorphism ϕ: F → F [X]/ and show that the translated polynomial ϕ(p) has a root over F [X]/. Because F is not a subfield of F [X]/ we construct in the second part the field (E \ ϕF )∪F for a given monomorphism ϕ : F → E and show that this field both is isomorphic to F and includes F as a subfield. In the literature this part of the proof usually consists of saying that “one can identify F with its image ϕF in F [X]/ and therefore consider F as a subfield of F [X]/”. Interestingly, to do so we need to assume that F ∩ E =∅, in particular Kronecker’s construction can be formalized for fields F with F \ F [X] =∅. Surprisingly, as we show in the third part, this condition is not automatically true for arbitray fields F : With the exception of 𝕑2 we construct for every field F an isomorphic copy F ′ of F with F′ ∩ F′ [X] ≠∅. We also prove that for Mizar’s representations of 𝕑n, 𝕈 and 𝕉 we have 𝕑n ∩ 𝕑n[X] = ∅, 𝕈 ∩ 𝕈[X] = ∅and 𝕉 ∩ 𝕉[X] = ∅, respectively. In the fourth part we finally define field extensions: E is a field extension of F i F is a subfield of E. Note, that in this case we have F ⊆ E as sets, and thus a polynomial p over F is also a polynomial over E. We then apply the construction of the second part to F [X]/ with the canonical monomorphism ϕ : F → F [X]/. Together with the first part this gives - for fields F with F ∩ F [X] = ∅ - a field extension E of F in which p ∈ F [X]\F has a root.

For a finite separable field extension K/k, all subfields can be obtained by intersecting so-called principal subfields of K/k. In this work we present a way to quickly compute these intersections. If the number of subfields is high, then this leads to faster run times and an improved complexity., Comment: Slides available at: http://www.math.fsu.edu/~hoeij/2017/Presentation.pdf

We prove a bound on the number of primes with a given splitting behavior in a given field extension. This bound generalizes the Brun–Titchmarsh bound on the number of primes in an arithmetic progression. The proof is set up as an application of Selberg’s Sieve in number fields. The main new ingredient is an explicit counting result estimating the number of integral elements with certain properties up to multiplication by units. As a consequence of this result, we deduce an explicit estimate for the number of ideals of norm up to [Formula: see text].

Let $p$ be a prime, let $K$ be a complete discrete valuation field of characteristic $0$ with a perfect residue field of characteristic $p$, and let $G_{K}$ be the Galois group. Let $\unicode[STIX]{x1D70B}$ be a fixed uniformizer of $K$, let $K_{\infty }$ be the extension by adjoining to $K$ a system of compatible $p^{n}$th roots of $\unicode[STIX]{x1D70B}$ for all $n$, and let $L$ be the Galois closure of $K_{\infty }$. Using these field extensions, Caruso constructs the $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D70F})$-modules, which classify $p$-adic Galois representations of $G_{K}$. In this paper, we study locally analytic vectors in some period rings with respect to the $p$-adic Lie group $\operatorname{Gal}(L/K)$, in the spirit of the work by Berger and Colmez. Using these locally analytic vectors, and using the classical overconvergent $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$-modules, we can establish the overconvergence property of the $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D70F})$-modules.

We present algorithmic, complexity and implementation results for the problem of isolating the real roots of a univariate polynomial in B α ∈ L [ y ] , where L = Q ( α ) is a simple algebraic extension of the rational numbers. We revisit two approaches for the problem. In the first approach, using resultant computations, we perform a reduction to a polynomial with integer coefficients and we deduce a bound of O ˜ B ( N 8 ) for isolating the real roots of B α , where N is an upper bound on all the quantities (degree and bitsize) of the input polynomials. The bound becomes O ˜ B ( N 7 ) if we use Pan's algorithm for isolating the real roots. In the second approach we isolate the real roots working directly on the polynomial of the input. We compute improved separation bounds for the roots and we prove that they are optimal, under mild assumptions. For isolating the real roots we consider a modified Sturm algorithm, and a modified version of descartes ' algorithm. For the former we prove a Boolean complexity bound of O ˜ B ( N 12 ) and for the latter a bound of O ˜ B ( N 5 ) . We present aggregate separation bounds and complexity results for isolating the real roots of all polynomials B α k , when α k runs over all the real conjugates of α. We show that we can isolate the real roots of all polynomials in O ˜ B ( N 5 ) . Finally, we implemented the algorithms in C as part of the core library of MATHEMATICA and we illustrate their efficiency over various data sets.

Let O be a differential graded (possibly colored) operad defined over rationals. Let us assume that there exists a zig-zag of quasi-isomorphisms connecting O ⊗ K to its cohomology , where K is any field extension of Q . We show that for a large class of such dg operads, a formality quasi-isomorphism for O exists and can be constructed recursively. Every step of our recursive procedure involves a solution of a finite dimensional linear system and it requires no explicit knowledge about the zig-zag of quasi-isomorphisms connecting O ⊗ K to its cohomology.

For any central simple algebra over a field F which contains a maximal subfield M with non-trivial automorphism group G = AutF(M), G is solvable if and only if the algebra contains a finite chain of subalgebras which are generalized cyclic algebras over their centers (field extensions of F) satisfying certain conditions. These subalgebras are related to a normal subseries of G. A crossed product algebra F is hence solvable if and only if it can be constructed out of such a finite chain of subalgebras. This result was stated for division crossed product algebras by Petit and overlaps with a similar result by Albert which, however, was not explicitly stated in these terms. In particular, every solvable crossed product division algebra is a generalized cyclic algebra over F.

An 1861 theorem of Hermite asserts that for every field extension E / F of degree 5 there exists an element of E whose minimal polynomial over F is of the form $$f(x) = x^5 + c_2 x^3 + c_4 x + c_5$$ for some $$c_2, c_4, c_5 \in F$$ . We give a new proof of this theorem using techniques of Galois cohomology, under a mild assumption on F.

We present a method of Grobner bases with respect to several term orderings and use it to obtain new results on multivariate dimension polynomials of inversive difference modules. Then we use the difference structure of the module of Kahler differentials associated with a finitely generated inversive difference field extension of a given difference transcendence degree to describe the form of a multivariate difference dimension polynomial of the extension.

Let A be a biquaternion division algebra over a field F ( char F ≠ 2 ). We prove that an element e ∈ F ⁎ such that A F ( e 4 ) = 0 exists if and only if there is a decomposition A ≃ ( a , b ) ⊗ ( c , d ) with 《 − a , b , c , d 》 = 0 . This statement strengthens the result of Rost, Serre and Tignol ( [7] ), where the additional condition − 1 ∈ F ⁎ holds. Another result is a construction of a field F with − 1 ∈ F ⁎ , a quartic cyclic field extension L / F , and quaternion algebras ( a , b ) and ( c , d ) over F such that ( a , b ) L = ( c , d ) L = 0 , but the quadratic form 〈 a , b , a b 〉 ⊥ 《 c , d 》 is anisotropic. As a consequence, for any algebraically closed field k 0 , we give an example of an anisotropic 7-dimensional quadratic form Ψ over k 0 ( z , x , y ) with ind ( C 0 ( Ψ ) ) = 4 such that for any discrete k 0 ( z ) -valuation v on k 0 ( z , x , y ) the form Ψ becomes isotropic over the completion k 0 ( z , x , y ) v . In other words, the form Ψ provides a counterexample to the strong Hasse principle for 7-dimensional quadratic forms with respect to the rational extension K ( x , y ) / K , where K = k 0 ( z ) .

We consider Hopf-Galois structures on separable (but not necessarily normal) field extensions $L/K$ of squarefree degree $n$. If $E/K$ is the normal closure of $L/K$ then $G=\mathrm{Gal}(E/K)$ can be viewed as a permutation group of degree $n$. We show that $G$ has derived length at most $4$, but that many permutation groups of squarefree degree and of derived length $2$ cannot occur. We then investigate in detail the case where $n=pq$ where $q \geq 3$ and $p=2q+1$ are both prime. (Thus $q$ is a Sophie Germain prime and $p$ is a safeprime). We list the permutation groups $G$ which can arise, and we enumerate the Hopf-Galois structures for each $G$. There are six such $G$ for which the corresponding field extensions $L/K$ admit Hopf-Galois structures of both possible types., Comment: 26 pages. Proposition 4.12 and Table 5 corrected. Theorem 3.5 removed. Typos fixed and some improvements made to wording. Accepted for Journal of Pure and Applied Algebra

We study GL_2(F)-invariant periods on representations of GL_2(A), where F is a nonarchimedean local field and A/F a product of field extensions of total degree 3. For irreducible representations, a theorem of Prasad shows that the space of such periods has dimension at most 1, and is non-zero when a certain epsilon-factor condition holds. We give an extension of this result to a certain class of reducible representations (of Whittaker type), extending results of Harris--Scholl when A is the split algebra F x F x F., Comment: Revised version, to appear in Forum Mathematicum

We give a geometric interpretation of Broussous-Grabitz embedding types. We fix a central division algebra $D$ of finite index over a non-Archimedean local field $F$ and a positive integer $m$. Further we fix a hereditary order $\mathfrak{a}$ of $\operatorname{M}_m(D)$ and an unramified field extension $E|F$ in $\operatorname{M}_m(D)$ which is embeddable in $D$ and which normalizes $\mathfrak{a}$. Such a pair $(E,\mathfrak{a})$ is called an embedding. The embedding types classify the $\operatorname{GL}_m(D)$-conjugation classes of these embeddings. Such a type is a class of matrices with non-negative integer entries. We give a formula which allows us to recover the embedding type of $(E,\mathfrak{a})$ from the simplicial type of the image of the barycenter of $\mathfrak{a}$ under the canonical isomorphism, from the set of $E^\times$-fixed points of the reduced building of $\operatorname{GL}_m(D)$ to the reduced building of the centralizer of $E^\times$ in $\operatorname{GL}_m(D)$. Conversely the formula allows to calculate the simplicial type up to cyclic permutation of the Coxeter diagram.

Let k be a non-archimedean local field with residual characteristic p. Let G be a connected reductive group over k that splits over a tamely ramified field extension of k. Suppose p does not divide the order of the Weyl group of G. Then we show that every smooth irreducible complex representation of G(k) contains an $\mathfrak{s}$-type of the form constructed by Kim and Yu and that every irreducible supercuspidal representation arises from Yu's construction. This improves an earlier result of Kim, which held only in characteristic zero and with a very large and ineffective bound on p. By contrast, our bound on p is explicit and tight, and our result holds in positive characteristic as well. Moreover, our approach is more explicit in extracting an input for Yu's construction from a given representation., Comment: 39 pages, accepted for publication in Annals of Mathematics

The objective of this study is to evaluate and de-risk the extension of the Kahmah Group, which is a newly identified carbonate extension in the Eastern part of XX field, and to assess and unlock further appraisal and development opportunities. Eastern field correlation was done by correlating the wells that has encountered Kahmah carbonate, the ones showing low Gamma Ray (GR) traces. Well correlation was done mainly on Petrophysical properties of three vertical penetrations that show distinct and abrupt change from high GR (Base Nahr Umr Shale and/or Mahwis sandstones) to low GR reflecting Kahmah carbonates. Based on the thicknesses of Kahmah based on the vertical well data and commercial oil seen in all wells in the vicinity, thickness and expected oil distribution maps were created. The main findings from the correlation are: 1) Kahmah reservoir is divided into two pays and two non-pays zones as seen in XX-1H1. 2) The reservoir is thickest around XX-1 and thins out toward YY-10 to the south and disappears completely in XX-5 to the north, portraying a wedge- shape. Kahmah reservoir varies in thickness, with thickest interval encountered in XX-1 with 11m. The reservoir thins out towards the edge closer to YY-10 confirming the wedge-like structure. A conceptual understanding has been incorporated in generating cross-sections portraying and proving the wedge-shape structure of Kahmah. Like any other new formation development, many uncertainties are associated with Kahmah development like formation extension, reservoir/fluid properties, pressure behaviour and others. The key observations from this project can be summarized as following: Different packages of reservoir. May be of different or similar propertiesExtension of Kahmah towards northern part of XX field is still highly uncertainContinuity of Kahmah Reservoir is uncertainReservoir pressure and communication with Mahwis formation need to be evaluatedReservoir and fluid properties also need to be evaluated

Stragglers, adversaries and colluding workers are some of the key problems affecting the performance of a distributed computing system. There have been many works in reducing the recovery threshold (i.e. minimum number of workers the master needs to wait, to compute the final output), while tackling adversaries and colluding workers for providing security and data privacy. These works generally consider datasets over arbitrary fields i.e. fields of characteristic both zero and prime. In this paper, we show that, for distributed computing problems over finite fields, performing the computations over an appropriately-sized extension field can improve the recovery threshold with a trade off only in computational complexity while preserving the privacy and security parameters. We show this for two schemes: (i) Lagrange coded computing scheme for evaluating an arbitrary multivariate polynomial over a dataset over finite fields, proposed in [Q. Yu, N. Raviv, J. So, and A. S. Avestimehr, “Lagrange coded computing: Optimal design for resiliency, security and privacy,” arXiv:1806.00939v3] and (ii) private secure matrix multiplication discussed in [M. Kim, and J. Lee, “Private Secure Coded Computation,” arXiv:1902.00167]. When a proper degree of field extension is chosen, the proposed coding schemes is applicable even in cases where the original schemes are not applicable because of insufficient number of workers or insufficient field size.

Gabidulin codes over fields of characteristic zero were recently constructed by Augot et al., whenever the Galois group of the underlying field extension is cyclic. In parallel, the interest in sparse generator matrices of Reed-Solomon and Gabidulin codes has increased lately, due to applications in distributed computations. In particular, a certain condition pertaining the intersection of zero entries at different rows, was shown to be necessary and sufficient for the existence of the sparsest possible generator matrix of Gabidulin codes over finite fields. In this paper we complete the picture by showing that the same condition is also necessary and sufficient for Gabidulin codes over fields of characteristic zero. Our proof builds upon and extends tools from the finite field case, combines them with a variant of the Schwartz-Zippel lemma over automorphisms, and provides a simple randomized construction algorithm whose probability of success can be arbitrarily close to one. In addition, potential applications for low-rank matrix recovery are discussed., Comment: ISIT'20

In the work short review of equipment applied for driving of underground utility lines and for hammering of vertical steel elements into soil is carried out. Construction and operating principle of pneumatic hammer with elastic circle and inertial valves for hammering of pipes with various diameter are described. Huge longitudinal forces generated during percussion provide rock destruction or soil compaction in the area of hole boring. The forces can be used for realization of other technological processes, for example, for repairing of oil hole electric-centrifugal pumps which require full disassemble. Application of the pneumatic hummer with mass of percussion part 40 kg has been probed for extraction many components (working wheels and guide vanes) from pipe barrel. Other example of percussion device application is intensification of unloading process of bulk materials from railway cars using vibro-percussion setup with drive from pneumodrift, change of which to analogous in size and working parameters allows to get high efficiency of the technological process.

We develop a modification of a notion of distance of an element in a valued field extension introduced by F.-V. Kuhlmann. We show that the new notion preserves the main properties of the distance and at the same time gives more complete information about a valued field extension. We study valued field extensions of prime degree to show the relation between the distances of the elements and the corresponding extensions of value groups and residue fields. In connection with questions related to defect extensions of valued function fields of positive characteristic, we present constructions of defect extensions of rational function fields K(x, y)|K generated by elements of various distances from K(x, y). In particular, we construct dependent Artin–Schreier defect extensions of K(x, y) of various distances.